p-Poincaré inequality versus ∞-Poincaré inequality: some counterexamples

We point out some of the differences between the consequences of p-Poincaré inequality and that of ∞-Poincaré inequality in the setting of doubling metric measure spaces. Based on the geometric characterization of ∞-Poincaré inequality given in Durand-Cartagena et al. (Mich Math J 60, 2011), we obtain a geometric property implied by the support of a p-Poincaré inequality, and demonstrate by examples that an analogous geometric characterization for finite p is not possible. The examples we give are metric measure spaces which are doubling and support an ∞-Poincaré inequality, but support no finite p-Poincaré inequality. In particular, these examples show that one cannot expect a self-improving property for ∞-Poincaré inequality in the spirit of Keith–Zhong (Ann Math 167(2):575–599, 2008). We also show that the persistence of Poincaré inequality under measured Gromov–Hausdorff limits fails for ∞-Poincaré inequality.     

On the composition and decomposition of clutters

The decomposition theory for simple n-person games is discussed in the context of clutters on a finite set. The notion of committee is defined, and an algorithm is given to find committees of any clutter. It is shown that this decomposition leads to a decomposition scheme for solving bottleneck and shortest path problems over clutters. Finally it is shown that the property of satisfying the length-width inequality is preserved under composition.     

A generalized real-valued measure of the inequality associated with a fuzzy random variable

Fuzzy random variables have been introduced by Puri and Ralescu as an extension of random sets. In this paper, we first introduce a real-valued generalized measure of the “relative variation” (or inequality) associated with a fuzzy random variable. This measure is inspired in Csiszár's f-divergence, and extends to fuzzy random variables many well-known inequality indices. To guarantee certain relevant properties of this measure, we have to distinguish two main families of measures which will be characterized. Then, the fundamental properties are derived, and an outstanding measure in each family is separately examined on the basis of an additive decomposition property and an additive decomposability one. Finally, two examples illustrate the application of the study in this paper.     

Time optimal boundary controls for the heat equation

The fact that the time optimal controls for parabolic equations have the bang–bang property has been recently proved for controls distributed inside the considered domain (interior control). The main result in this article asserts that the boundary controls for the heat equation have the same property, at least in rectangular domains. This result is proved by combining methods from traditionally distinct fields: the Lebeau–Robbiano strategy for null controllability and estimates of the controllability cost in small time for parabolic systems, on one side, and a Remez-type inequality for Müntz spaces and a generalization of Turán?s inequality, on the other side.     

Reducible means and reducible inequalities

It is well-known that if a real valued function acting on a convex set satisfies the n-variable Jensen inequality, for some natural number \(n\ge 2\), then, for all \(k\in \{1,\dots , n\}\), it fulfills the k-variable Jensen inequality as well. In other words, the arithmetic mean and the Jensen inequality (as a convexity property) are both reducible. Motivated by this phenomenon, we investigate this property concerning more general means and convexity notions. We introduce a wide class of means which generalize the well-known means for arbitrary linear spaces and enjoy a so-called reducibility property. Finally, we give a sufficient condition for the reducibility of the (M, N)-convexity property of functions and also for Hölder–Minkowski type inequalities.     

Nulle contrôlabilité régionale d'une équation de type Crocco linéariséeRegional null controllability of a linearized Crocco type equation

We are interested in controllability problems of equations coming from a boundary layer model. This problem is described by a degenerate parabolic equation (a linearized Crocco type equation) where phenomena of diffusion and transport are coupled.First we give a geometric characterization of the influence domain of a locally distributed control. Then we prove regional null controllability results on this domain. The proof is based on an adequate observability inequality for the homogeneous adjoint problem. This inequality is obtained by decomposition of the space–time domain and Carleman type estimates along characteristics. To cite this article: P. Martinez et al., C. R. Acad. Sci. Paris, Ser. I 334 (2002) 581–584.     

Classes of tuples of commuting contractions satisfying the multivariable von Neumann inequality

We obtain a decomposition for multivariable Schur-class functions on the unit polydisk which, to a certain extent, is analogous to Agler's decomposition for functions from the Schur-Agler class. As a consequence, we show that d-tuples of commuting strict contractions obeying an additional positivity constraint satisfy the d-variable von Neumann inequality for an arbitrary operator-valued bounded analytic function on the polydisk. Also, this decomposition yields a necessary condition for solvability of the finite data Nevanlinna-Pick interpolation problem in the Schur class on the unit polydisk.     

Algorithme de Neumann–Dirichlet pour des problèmes de contact unilatéral : Résultat de convergenceNeumann–Dirichlet algorithm for unilateral contact problems: Convergence results

In this Note, we propose and we prove the convergence of a Neumann–Dirichlet algorithm in order to approximate a Signorini problem between two elastic bodies. The idea is to retain the natural interface between the two bodies as numerical interface for the domain decomposition and to replace the Dirichlet problem in [4] by a variational inequality. To cite this article: G. Bayada et al., C. R. Acad. Sci. Paris, Ser. I 335 (2002) 381–386.     

Équations de Hamilton–Jacobi et inégalités entropiques généraliséesHamilton–Jacobi equations and inequalities for generalised entropies

We prove the equivalence between a general logarithmic Sobolev inequality and the hypercontractivity of a Hamilton–Jacobi equation. We also recover that this property imply a transportation inequality established by [5]. These results provide a natural generalization of the work performed in [3]. To cite this article: I. Gentil, F. Malrieu, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 437–440.     

Eigenvalue inequalities for the buckling problem of the drifting Laplacian

In this paper, we study the buckling problem of the drifting Laplacian on bounded domains in a complete Riemannian manifold with nonnegative ∞-dimensional Bakry–Émery Ricci curvature. According to the property of the manifold, we obtain a family of trial functions. By making use of these trial functions, we derive a universal inequality of eigenvalues, which is independent of the domains.     

The Riesz Basis Property of an Indefinite Sturm–Liouville Problem with Non-Separated Boundary Conditions

We consider a regular indefinite Sturm–Liouville eigenvalue problem ?f′′ + q f = λ r f on [a, b] subject to general self-adjoint boundary conditions and with a weight function r which changes its sign at finitely many, so-called turning points. We give sufficient and in some cases necessary and sufficient conditions for the Riesz basis property of this eigenvalue problem. In the case of separated boundary conditions we extend the class of weight functions r for which the Riesz basis property can be completely characterized in terms of the local behavior of r in a neighborhood of the turning points. We identify a class of non-separated boundary conditions for which, in addition to the local behavior of r in a neighborhood of the turning points, local conditions on r near the boundary are needed for the Riesz basis property. As an application, it is shown that the Riesz basis property for the periodic boundary conditions is closely related to a regular HELP-type inequality without boundary conditions.     

On the root mean square weighted L2 discrepancy of scrambled nets

Until now (t,m,s)-nets in base b are the most important representatives in the family of low-discrepancy point sets. Such nets are often used for quasi-Monte Carlo approximation of high-dimensional integrals. Owen introduced a randomization of such point sets such that the net property is preserved. In this paper we consider the root mean square weighted L₂ discrepancy of (0,m,s)-nets in base b. The concept of weighted discrepancy was introduced by Sloan and Woźniakowski to give a general form of a Koksma–Hlawka inequality that takes into account imbalances in the “importance” of the projections of the integrand.     

Levitin–Polyak well-posedness by perturbations of inverse variational inequalities

The purpose of this paper is to investigate Levitin–Polyak type well-posedness for inverse variational inequalities. We establish some metric characterizations of Levitin–Polyak α-well-posedness by perturbations. Under suitable conditions, we prove that Levitin–Polyak well-posedness by perturbations of an inverse variational inequality is equivalent to the existence and uniqueness of its solution. Moreover, we show that Levitin–Polyak well-posedness by perturbations of an inverse variational inequality is equivalent to Levitin–Polyak well-posedness by perturbations of an enlarged classical variational inequality.     

A Note on Weighted Korn Inequality

In this note, we show that, for domains satisfying the separation property, certain weighted Korn inequality is equivalent to the John condition. Our result generalizes previous result from Jiang–Kauranen [Calc. Var. Partial Differential Equations, 56, Art. 109,(2017)] to weighted settings.     

Perturbation of the CR fractional Yamabe problem

In this note, we first prove the non‐degeneracy property of extremals for the optimal Hardy–Littlewood–Sobolev inequality on the Heisenberg group, as an application, a perturbation result for the CR fractional Yamabe problem is obtained, this generalizes a classical result of Malchiodi and Uguzzoni 30 .     

Stability of isoperimetric type inequalities for some Monge–Ampère functionals

We investigate the stability of some inequalities of isoperimetric type related to Monge–Ampère functionals. In particular, firstly we prove the stability of a reverse Faber–Krahn inequality for the Monge–Ampère eigenvalue and its generalization. Then we give a stability result for the Brunn–Minkowski inequality and for a consequent Urysohn’s type inequality for the so-called \(n\) -torsional rigidity, a natural extension of the usual torsional rigidity.     

Subadditivity of The Entropy and its Relation to Brascamp–Lieb Type Inequalities

We prove a general duality result showing that a Brascamp–Lieb type inequality is equivalent to an inequality expressing subadditivity of the entropy, with a complete correspondence of best constants and cases of equality. This opens a new approach to the proof of Brascamp–Lieb type inequalities, via subadditivity of the entropy. We illustrate the utility of this approach by proving a general inequality expressing the subadditivity property of the entropy on ${\mathbb {R}^n}We prove a general duality result showing that a Brascamp–Lieb type inequality is equivalent to an inequality expressing subadditivity of the entropy, with a complete correspondence of best constants and cases of equality. This opens a new approach to the proof of Brascamp–Lieb type inequalities, via subadditivity of the entropy. We illustrate the utility of this approach by proving a general inequality expressing the subadditivity property of the entropy on \mathbb Rⁿ{\mathbb {R}^n}, and fully determining the cases of equality. As a consequence of the duality mentioned above, we obtain a simple new proof of the classical Brascamp–Lieb inequality, and also a fully explicit determination of all of the cases of equality. We also deduce several other consequences of the general subadditivity inequality, including a generalization of Hadamard’s inequality for determinants. Finally, we also prove a second duality theorem relating superadditivity of the Fisher information and a sharp convolution type inequality for the fundamental eigenvalues of Schr?dinger operators. Though we focus mainly on the case of random variables in \mathbb Rⁿ{\mathbb {R}^n} in this paper, we discuss extensions to other settings as well.     

One-cohomology and the uniqueness of the group measure space decomposition of a II1 factor

We provide a unified and self-contained treatment of several of the recent uniqueness theorems for the group measure space decomposition of a II₁ factor. We single out a large class of groups Γ, characterized by a one-cohomology property, and prove that for every free ergodic probability measure preserving action of Γ the associated II₁ factor has a unique group measure space Cartan subalgebra up to unitary conjugacy. Our methods follow closely a recent article of Chifan–Peterson, but we replace the usage of Peterson’s unbounded derivations by Thomas Sinclair’s dilation into a malleable deformation by a one-parameter group of automorphisms.     

Universal inequalities for lower order eigenvalues of self-adjoint operators and the poly-Laplacian

In this paper, we first establish an abstract inequality for lower order eigenvalues of a self-adjoint operator on a Hilbert space which generalizes and extends the recent results of Cheng et al. （Calc. Var. Partial Differential Equations, 38, 409-416 （2010））. Then, making use of it, we obtain some universal inequalities for lower order eigenvalues of the biharmonic operator on manifolds admitting some speciM functions. Moreover, we derive a universal inequality for lower order eigenvalues of the poly-Laplacian with any order on the Euclidean space.     

Functional calculus and duality for closed operators

In this paper, we continue our spectral-theoretic study [8] of unbounded closed operators in the framework of the spectral decomposition property and decomposable operators. Given a closed operator T with nonempty resolvent set, let f → f(T) be the homomorphism of the functional calculus. We show that if T has the spectral decomposition property, then f(T) is decomposable. Conversely, if f is nonconstant on every component of its domain which intersects the spectrum of T, then f(T) decomposable implies that T has the spectral decomposition property. A spectral duality theorems follows as a corollary. Furthermore, we obtain an analytic-type property for the canonical embedding J of the underlying Banach space X into its second dual $\text{X}{}^7$.